Positivity of the generalized translation associated with the \(q\)-Hankel transform
type: Item
instance of: scholarly article
title: Positivity of the generalized translation associated with the \(q\)-Hankel transform
zbMATH Open document ID: 1247.33030
author: Ahmed Fitouhi
author: Lazhar Dhaouadi
published in: Constructive Approximation
publication date: 2012-02-13T00:00:00Z
review text: The authors study the positivity of a \(q\)-analogue of the Hankel translation \[ T_x^{\nu}f(y)=\int_0^{\infty}D_{\nu}(x,y,t)f(t)\,dt, \] where \[ D_{\nu}(x,y,z)=\int_0^{\infty}t^{-\nu-1/2}(xt)^{1/2}J_{\nu}(xt)(y)^{1/2}J_{\nu}(yt)(zt)^{1/2}J_{\nu}(zt)\,dt. \] This \(q\)-analogue is defined in terms of the normalized third Jackson \(q\)-Bessel function and its positivity was already posed in [\textit{L.~Dhaouadi, A.~Fitouhi} and \textit{J.~El~Karmel}, JIPAM, J. Inequal. Pure Appl. Math. 7, No. 5, Paper No. 171, 14 p. (2006; Zbl 1232.26025)]. For the positivity Graf's addition formula for the third Jackson \(q\)-Bessel function plays a central role. This addition formula was proved in [\textit{H.~T. Koelink} and \textit{R.~F.~Swarttouw}, J. Approximation Theory 81, No. 2, 260--273 (1995; Zbl 0821.33010)].
reviewed by: Roelof Koekoek
Mathematics Subject Classification ID: 33D15
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 6005983
zbMATH Keywords: \(q\)-Bessel transform
zbMATH Keywords: \(q\)-Hankel transform
zbMATH Keywords: Graf's addition formula
MaRDI profile type: Publication
full work available at URL: s00365-011-9132-0
OpenAlex ID: W2077494136
cites work: Elements of harmonic analysis related to the third basic zero order Bessel function
cites work: The Weierstrass Hankel convolution transform
cites work: Q5433822
cites work: The \textit{q-j}\(_\alpha\) Bessel function
cites work: Q4828543
cites work: On q-Analogues of the Fourier and Hankel Transforms
cites work: The quantum group of plane motions and the Hahn-Exton \(q\)-Bessel function
cites work: A \(q\)-analogue of Graf's addition formula for the Hahn-Exton \(q\)-Bessel function
cites work: An Addition Theorem and Some Product Formulas for the Hahn-Exton q-Bessel Functions
cites work: Q5562752
DOI: 10.1007/S00365-011-9132-0
description: scientific article; zbMATH DE number 6005983